View source for Lattice-ordered group

<!--
l0576702.png
$#A+1 = 80 n = 0
$#C+1 = 80 : ~/encyclopedia/old_files/data/L057/L.0507670 Lattice\AAhordered group,
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

'' $  l $-group''

A [[Group|group]]  $  G $
on the set of elements of which a partial-order relation  $  \leq  $
is defined possessing the properties: 1)  $  G $
is a [[Lattice|lattice]] relative to  $  \leq  $, 
i.e. for any  $  x, y \in G $
there are elements  $  x \wedge y $, 
$  x \lor y $
such that  $  x \wedge y \leq  x, y $
and  $  x \lor y \geq  x, y $; 
for any  $  z \in G $, 
$  z \leq  x, y $
implies  $  z \leq  x \wedge y $, 
and for any  $  t \in G $
and  $  x, y \leq  t $
one has  $  x \lor y \leq  t $; 
and 2) for any  $  a, b, x, y \in G $
the inequality  $  a \leq  b $
implies  $  xay \leq  xby $. 
Similarly, a lattice-ordered group can be defined as an [[Algebraic system|algebraic system]] of signature  $  \langle  \cdot , {}  ^ {- 1} , e, \wedge, \lor \rangle $
that satisfies the axioms: 3)  $  \langle  G, \cdot , {}  ^ {- 1} , e\rangle $
is a group; 4)  $  \langle  G, \lor , \wedge\rangle $
is a lattice; and 5)  $  x( y \lor z) t = xyt \lor xzt $
and  $  x( y \wedge z) t = xyt \wedge xzt $
for any  $  x, y, z, t \in G $.

The lattice of elements of a lattice-ordered group is distributive (cf. [[Distributive lattice|Distributive lattice]]). The absolute value (respectively, the positive and the negative part) of an element  $  x $
is the element  $  | x | = x \lor x  ^ {- 1} $ (respectively,  $  x  ^ {+} = x\lor e $
and  $  x  ^ {-} = x \wedge e $). 
In lattice-ordered groups, the following relations hold:

$$ 
x  =  x  ^ {+} x  ^ {-} ,\ \ 
| x |  ^ {- 1}  \leq   x  \leq   | x | ,
$$

$$ 
| x |  =  x  ^ {+} ( x  ^ {-} )  ^ {- 1} ,\  x  ^ {+} \wedge ( x  ^ {-} )  ^ {- 1 } =  e,
$$

$$ 
( x \lor y)  ^ {- 1}  =  x  ^ {- 1} \wedge y  ^ {- 1} ,\ \ 
( x \wedge y)  ^ {- 1}  =  x  ^ {- 1} \lor y  ^ {- 1} .
$$

Two elements  $  x $
and  $  y $
are called orthogonal if  $  | x | \lor | y | = e $. 
Orthogonal elements commute.

A subset  $  H $
of an  $  l $-group  $  G $
is called an  $  l $-subgroup if  $  H $
is a subgroup and a sublattice in  $  G $; 
an  $  l $-subgroup  $  H $
is called an  $  l $-ideal of  $  G $
if it is normal and convex in  $  G $. 
The set of  $  l $-subgroups of a lattice-ordered group forms a sublattice of the lattice of all its subgroups. The lattice of  $  l $-ideals of a lattice-ordered group is distributive. An  $  l $-homomorphism of an  $  l $-group  $  G $
into an  $  l $-group  $  H $
is a [[Homomorphism|homomorphism]]  $  \phi $
of the group  $  G $
into the group  $  H $
such that

$$ 
\phi ( x \lor y)  =  \phi ( x) \lor \phi ( y) ,\ \ 
\phi ( x \wedge y)  =  \phi ( x) \wedge \phi ( y).
$$

The kernels of  $  l $-homomorphisms are precisely the  $  l $-ideals of  $  l $-groups. If  $  G $
is an  $  l $-group and  $  M \subset  G $, 
then the set  $  M  ^  \perp  = \{ {x \in G } : {| x | \wedge | m | = e  \textrm{ for  every  }   m \in M } \} $
is a convex  $  l $-subgroup in  $  G $ (cf. [[Convex subgroup|Convex subgroup]]).

The group  $  A( L) $
of one-to-one order-preserving mappings of a totally ordered set  $  L $
onto itself is an  $  l $-group (if for  $  f, g \in A( L) $
one assumes that  $  f \leq  g $
if and only if  $  f( \alpha ) \leq  g( \alpha ) $
for all  $  \alpha \in L $). 
Every  $  l $-group is  $  l $-isomorphic to an  $  l $-subgroup of the lattice-ordered group  $  A( L) $
for a suitable set  $  L $.

The class of all lattice-ordered groups is a variety of signature  $  \langle  \cdot , {}  ^ {- 1} , e, \wedge, \lor\rangle $ (cf. [[Variety of groups|Variety of groups]]). Its most important subvariety is the class of lattice-ordered groups that can be approximated by totally ordered groups (the class of representable  $  l $-groups, cf. also [[Totally ordered group|Totally ordered group]]).

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  G. Birkhoff,   "Lattice theory" , ''Colloq. Publ.'' , '''25''' , Amer. Math. Soc.  (1973)</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  L. Fuchs,   "Partially ordered algebraic systems" , Pergamon  (1963)</TD></TR>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Anderson,   T. Feil,   "Lattice-ordered groups. An introduction" , Reidel  (1988)</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  A.M.W. Glass (ed.)  W.Ch. Holland (ed.) , ''Lattice-ordered groups. Advances and techniques'' , Kluwer  (1989)</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Martinez (ed.) , ''Ordered algebraic structures'' , Kluwer  (1989)</TD></TR>
</table>